Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  suctrALT2VD Structured version   Visualization version   GIF version

Theorem suctrALT2VD 38593
Description: Virtual deduction proof of suctrALT2 38594. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
suctrALT2VD (Tr 𝐴 → Tr suc 𝐴)

Proof of Theorem suctrALT2VD
Dummy variables 𝑧 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dftr2 4724 . . 3 (Tr suc 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
2 sssucid 5771 . . . . . . . 8 𝐴 ⊆ suc 𝐴
3 idn1 38311 . . . . . . . . 9 (   Tr 𝐴   ▶   Tr 𝐴   )
4 idn2 38359 . . . . . . . . . 10 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   (𝑧𝑦𝑦 ∈ suc 𝐴)   )
5 simpl 473 . . . . . . . . . 10 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧𝑦)
64, 5e2 38377 . . . . . . . . 9 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   𝑧𝑦   )
7 idn3 38361 . . . . . . . . 9 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦𝐴   ▶   𝑦𝐴   )
8 trel 4729 . . . . . . . . . 10 (Tr 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
98expd 452 . . . . . . . . 9 (Tr 𝐴 → (𝑧𝑦 → (𝑦𝐴𝑧𝐴)))
103, 6, 7, 9e123 38510 . . . . . . . 8 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦𝐴   ▶   𝑧𝐴   )
11 ssel 3582 . . . . . . . 8 (𝐴 ⊆ suc 𝐴 → (𝑧𝐴𝑧 ∈ suc 𝐴))
122, 10, 11e03 38488 . . . . . . 7 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦𝐴   ▶   𝑧 ∈ suc 𝐴   )
1312in3 38355 . . . . . 6 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   (𝑦𝐴𝑧 ∈ suc 𝐴)   )
14 idn3 38361 . . . . . . . . 9 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦 = 𝐴   ▶   𝑦 = 𝐴   )
15 eleq2 2687 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝑧𝑦𝑧𝐴))
1615biimpcd 239 . . . . . . . . 9 (𝑧𝑦 → (𝑦 = 𝐴𝑧𝐴))
176, 14, 16e23 38503 . . . . . . . 8 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦 = 𝐴   ▶   𝑧𝐴   )
182, 17, 11e03 38488 . . . . . . 7 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦 = 𝐴   ▶   𝑧 ∈ suc 𝐴   )
1918in3 38355 . . . . . 6 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   (𝑦 = 𝐴𝑧 ∈ suc 𝐴)   )
20 simpr 477 . . . . . . . 8 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑦 ∈ suc 𝐴)
214, 20e2 38377 . . . . . . 7 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   𝑦 ∈ suc 𝐴   )
22 elsuci 5760 . . . . . . 7 (𝑦 ∈ suc 𝐴 → (𝑦𝐴𝑦 = 𝐴))
2321, 22e2 38377 . . . . . 6 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   (𝑦𝐴𝑦 = 𝐴)   )
24 jao 534 . . . . . 6 ((𝑦𝐴𝑧 ∈ suc 𝐴) → ((𝑦 = 𝐴𝑧 ∈ suc 𝐴) → ((𝑦𝐴𝑦 = 𝐴) → 𝑧 ∈ suc 𝐴)))
2513, 19, 23, 24e222 38382 . . . . 5 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   𝑧 ∈ suc 𝐴   )
2625in2 38351 . . . 4 (   Tr 𝐴   ▶   ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴)   )
2726gen12 38364 . . 3 (   Tr 𝐴   ▶   𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴)   )
28 biimpr 210 . . 3 ((Tr suc 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴)) → (∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴) → Tr suc 𝐴))
291, 27, 28e01 38437 . 2 (   Tr 𝐴   ▶   Tr suc 𝐴   )
3029in1 38308 1 (Tr 𝐴 → Tr suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384  wal 1478   = wceq 1480  wcel 1987  wss 3560  Tr wtr 4722  suc csuc 5694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3192  df-un 3565  df-in 3567  df-ss 3574  df-sn 4156  df-uni 4410  df-tr 4723  df-suc 5698  df-vd1 38307  df-vd2 38315  df-vd3 38327
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator